Supplementary Online Material For

Supplementary Online Material for

Analyzing the Distribution of Threshold Voltage Degradation in Nanoscale Transistors by Using Reaction-Diffusion and Percolation Theory

*Ahmad Ehteshamul Islam and **Muhammad Ashraful Alam

*Department of Materials Science and Engineering, University of Illinois, Urbana, IL61801, USA

**School of Electrical and Computer Engineering, PurdueUniversity, West Lafayette, IN47907-2035, USA.

Email: *; **

We prepare the following supplementary material to facilitate the discussion in section 2.2 of the main manuscript. Consequently, we explain the underlying physics of interface defect distribution at fixed stress time, f(NIT)@tSTS, and stress time distribution at fixed amount of interface defect, f(tSTS)@NIT. Though atomic hydrogen (H)-diffusion is not relevant for NBTI analysis, we begin with the reformulation of H-only R-D model in sectionS1 by defining Markov-Chain transition matrix. The analytical/numerical Markov Chain Monte-Carlo (MCMC) solution of the H-only R-D model (sections S2-S3) allows us to highlight some of the general statistical features of the R-D model. Since the statistics of the R-D model with H-H2 diffusion (presented in section 2.2 of the main manuscript) follows the general features of the H-only R-D model (presented in this supplementary material), the physical explanation of the R-D statistics, provided in this supplementary material, should be valid for any R-D system, in general.

S1. Transition Matrix for R-D Model with H Diffusion

The rate equations governing the interface defect generation in the R-D model have the following form[1]:

(1)

(2)

(3)

Equation(1) represents passivation/de-passivation effects of Si-H bond, where kF, kR, N0, NIT, NH(0) are defined as Si-H bond-breaking rate, Si-H bond-annealing rate, initial bond density available before stress, interface defect density per unit area of MOSFET and atomic hydrogen density per unit volume at the Si/dielectric interface, respectively. Equation(2) describes diffusion (along x axis) of H, whereas eqn. (3) corresponds to the conservation of forward and reverse fluxes of diffusing H near the interface. In eqns. (2) and (3), DH, NH, and δ represent H diffusion coefficient, H concentration, and the interfacial thickness, respectively.

To create the transition matrix for H-diffusion based R-D system, we first discretize eqns. (1)-(3) for constant time step Δt and constant grid size Δxalong the diffusion direction [2,3]. A rearrangement of the discretized version of R-D equations results –

(4)

(5)

(6)

where pF = kFΔt, pR = kRNITΔt/Δx, and = DHΔtΔx-2. Equations(4)-(6) enable us to define the Markov model of Fig. S1, which is later used to analytically and numerically study the interface defect statistics using R-D theory. We can also express eqns. (4)-(6) in the following matrix form:

(7)

In eqn. (7), [M]i is the Markov model transition matrix defining the transition and staying probability of one Monte-Carlo particle after the i-th time step, {ρ}i = [(N0-NIT,i)/Δx NH,ij=0 NH,ij=1 …]T is a vector for storing the interface defect density and hydrogen concentration at different spatial grid points j at the i-th time step. The diagonal elements of [M]i indicate the ‘staying’ probability of each Monte-Carlo particle; whereas the off-diagonal elements indicate the ‘transition’ probabilities in forward or reverse direction within the simulation domain. It is to be noted here that eqn. (7) is similar to the one developed in Ref. [3], except [M]i was considered as time-invariant in Ref. [3]. Since pR is time-dependent for interface defect generation, [M]i is time-varying for the problem under study.

After defining the matrix equation for the R-D model having H-diffusion, we determine the average value of NIT and NH j at different tSTS = [t1 t2 t3 …] using eqn. (7). The calculation starts with the definition of ρ(tSTS = 0) = {ρ}0 = [N0/Δx 0 0 …]T, followed by the use of {ρ}i = [M] i-1{ρ}i-1 tofind the time evolution of mean {ρ}i at different ti. Next, we perform Monte-Carlo simulation to determine the statistics of {ρ}i for i > 0. During Monte-Carlo simulation, we use {ρ}0 = [N0LW 0 0 …]T(where L and W are the channel length and width of the transistor under consideration) and then use the transition and staying probabilities of eqn. (7) at each time step ti to determine the overall statistics.

S2. Statistics of R-D System with H Diffusion

The transition matrix of section S1 is used here to obtain the statistics of interface defect. We analytically (see sections S2.1 and S2.2) and then numerically (see section S2.3) solve the time evolution of interface defect’s statistics. The statistical features of R-D system with H-diffusion, presented in this section, is similar to the one obtained for H-H2 diffusion in section 2.2 of the main manuscript.

S2.1. NIT Statistics for H Diffusion: Analytical Calculation for f(tSTS)@NIT

Power series generating functions [4] are well adapted to calculate the moments of f(tSTS)@NIT by using the transition and staying probabilities of eqn. (7). It can be shown using generating function that the mean stress time for a particular amount of interface defect is [5-7]:

(8)

where p’s and q’s are the right (moving away from the Si/dielectric interface in the R-D system under study) and left (moving towards the Si/dielectric interface) transition probabilities, respectively; S is the average number of Monte-Carlo steps taken for a particular amount of interface defect. For example, S = 0 means no Si-H bond dissociation, S = 1 means Si-H bond dissociation with insignificant diffusion, and S > 1 means that the dissociated H has diffused (on average) a distance (S-1)Δx away from the interface. For eqn. (7), p0 = pF= kFΔt, q1 = pR = kRNITΔt/Δx, and p1 = p2 = p3 = ……= pS-1 =q2 = q3 = ……= qS == DHΔtΔx-2. Hence, using eqn. (8), we calculate the following expression for the mean of f(tSTS)@NIT:

(9)

Similarly, using generating function[4], we can also calculate the following expression for the variance of tSTS@NIT distribution:

(10)

In the reaction (when, S ~ 0) and diffusion (when, S 1) limits, eqns. (9)-(10) reduces to –

(11)

(12)

Here, eqn. (11) reflects that if there were no diffusion or annealing (i.e., both DH and kR are negligible), Si-H bond dissociation will occur after a mean time of ~1/kF, having a standard deviation of the same order. Interestingly, eqns. (11) and (12) suggest that for both the reaction- and diffusion-limited R-D system, . This is a general characteristic of the log-normal distribution having probability distribution function (PDF):

(13)

where ~ 1. For f(tSTS)@NIT, we find (or 0.362 in the base-10 log-scale), i.e., . Therefore,; in other words, our generating function calculation anticipates a log-normal distribution for f(tSTS)@NITfor any interface defect density, with an approximately constant σ. Obviously, for large stress time or for large-area transistors, when µ0,S is very large, the log-normal distribution of eqn. (13) reverts back to the well-known Gaussian or normal distribution.

S2.2. NIT Statistics for H Diffusion: Graphical Analysis of f(NIT)@tSTS

A graphical inversion of f(tSTS)@NIT allows us to determine the characteristics of f(NIT)@tSTS (see Fig. S2) that is regularly reported in NBTI statistics literature[8-13]. R-D system with H-diffusion has a power-law time exponent of n ~ 1 in the reaction regime and n ~ ¼ in the diffusion regime[1]. Such power-law solution suggests that when f(tSTS)@NIT is log-normal (following eqn. (13))with standard deviation σ, f(NIT)@tSTS should also be log-normal with standard deviation nσ. Thus, f(NIT)@tSTS will have σ ~0.362 (in base-10 log-scale) in the reaction regime and σ~0.0905 (also in base-10 log-scale) in the H-diffusion regime.

S2.3. NIT Statistics for H Diffusion: Numerical MCMC Simulation Results

Similar to section 2.2 of the main manuscript, the stochastic MCMC results are first compared with the continuum solution of eqns. (1)-(3) for a large (L = 100nm; W = 1um, Fig. S3a) and small (L = 50nm; W = 100nm, Fig. S3b) transistors. We also compute f(tSTS)@NIT and observe (see Fig. S4) that the distribution is symmetric around the median value in a semilog-x plot, while only being cut-off at both ends by the sampling time and the duration of simulation, respectively. These numerical solutions suggest that – (i) f(tSTS)@NIT follows a log-normal distribution and (ii) a constant σ ~ 0.5 describesf(tSTS)@NIT at all tSTS; both observations are consistent with our expectations, as described in sections S2.1 and S2.2.

While analyzing the statistics of f(NIT)@tSTS, we observe that f(NIT)@tSTSis clearly Gaussian at long stress time for the large transistor;however, there is significant skewness for f(NIT)@tSTSat small stress time (see Fig. S3a). Mean of f(NIT)@tSTS or µITin small stress time regime is much less than 1010 cm-2. Suchsmall value of µIT corresponds to a relatively small ΔVT,mean of < 1mV for a transistor with EOT ~ 1nm, which may not be detectable using conventional NBTI setup. On the other hand, forthe small transistor, f(NIT)@tSTS never approaches a Gaussian distribution, even at µIT ~2x1011 cm-2 (see Fig. S3b). Yet this level of interface defect is typical for NBTI experiments[14]. In addition, the standard deviation of f(NIT)@tSTS or σIT increases with µIT and the distribution has significant positive skewness, γIT (see Fig. S5). Thus, numerical MCMC analysis suggests the use of skewed Gaussian distribution[15] as the best fit for f(NIT)@tSTS in small-size or nanoscale transistors, i.e.,

(14)

where φ(x) and Ф(x) indicate the probability distribution function (PDF) and cumulative distribution function (CDF) of a Gaussian distribution, and ξIT, ωIT, αIT have the following relationship with the µIT, σIT, and γIT of f(NIT)@tSTS[15]:

(15)

To compare the goodness of fit between eqn. (14)and MCMC-simulated f(NIT)@tSTS, we first estimate the moments µIT, σIT, and γIT for the computed f(NIT)@tSTS. Then, we use eqn. (15) to determine the skew-normal distribution parameters ξIT, ωIT, and αIT, and then use eqn. (14) to fit the simulated f(NIT)@tSTS. Fig. S6 compares the MCMC-simulated f(NIT)@tSTS (obtained from 10000 MCMC simulations) in the small L = 50nm; W = 100nm transistor with the one obtained using eqns. (14)-(15) and they show an excellent match. The figure also fits f(NIT)@tSTS using Gaussian, Poisson, and log-normal distributions and therefore, suggests skew-normal distribution to have the least sum of squares (SSE).

S3. Comparison of Analytical/Graphical and MCMC Results

Although our analytical/graphical interpretation suggests log-normal distribution for both f(NIT)@tSTS and f(tSTS)@NIT(see Fig. S2 and sections S2.1 and S2.2), the numerical MCMC simulation suggests a log-normal behavior only for f(tSTS)@NIT (see Fig. S4), but not for f(NIT)@tSTS (see Fig. S3). This can be explained as follows: At long tSTS (diffusion regime) and for large transistor, σ of eqn. (13) for f(NIT)@tSTS will be 4 times smaller than the σ for f(tSTS)@NIT (see Fig. S2). Therefore, f(NIT)@tSTSwith such small σ in the diffusion regime will have a normal distribution, rather than a log-normal distribution. In addition, there is a constraint of NIT > 0 that one always need to satisfy when µIT is small. Such constraint of NIT > 0 makes the normal distribution right-skewed for small µIT. Existence of small µIT happens for a large transistor at very short tSTS and for a small transistor at any tSTS. Therefore, though log-normal distribution is expected for f(NIT)@tSTS, according to Fig. S2, due to the – (a) presence of small σ, f(NIT)@tSTS approaches a normal distribution at larger µIT, (b) constraint of NIT > 0, f(NIT)@tSTS approaches a skew-normal distribution for smaller µIT.

S4. Summary

The statistical analysis of the H-only R-D model, presented in this supplementary online material, enables us to identify the following features of R-D model:

• f(tSTS)@NIT will always be log-normal for any amount of interface defect. The distribution, fitted using eqn. (13), will have constant σ.
• f(NIT)@tSTSis skew-normal for smaller amount of interface defect, and approaches towards a normal distribution for a larger quantity.
• The standard deviation of f(NIT)@tSTS always increases withthe mean of f(NIT)@tSTS.

Since we observe the same features, as stated above, for the R-D system having H-H2 diffusion, as presented in section 2.2 of the main manuscript, the detailed mathematical calculation performed in this supplementary material justifies the use of skew-normal distribution for fitting f(NIT)@tSTS obtained using the R-D system having H-H2 diffusion.

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Fig. S1 Markov model for probabilistic motion in a R-D system having H diffusion

Fig. S2Schematic to illustrate the idea that a log-normal distribution for f(tSTS)@NIT guarantees a log-normal distribution for f(NIT)@tSTS with constant σ for the distribution expressed in the form of eqn. (13)

Fig. S3Time dependence and PDF of interface defect generation for (a) a large transistor (L = 100nm, W = 1um) and (b) a small transistor (L = 50nm, W = 100nm) using N0 = 5x1012 cm-2, kF = 0.1 sec-1, kR = 2x10-16 cm-3sec-1, and DH = 10-13 cm2-sec-1. For the large transistor, the PDF evolves from skewed normal (in the reaction phase) to a normal (in the diffusion phase) distribution. However, for the small transistor, the distribution stays skewed normal, even in the diffusion phase, which is commonly captured within the measurement window of NBTI measurement scheme

Fig.S4 Distribution of tSTS, f(tSTS) at different levels of NIT obtained from 1000 MCMC simulation in the small transistor under study. The distribution is clearly log-normal, having cut-off at tSTS ~ 0.1 sec (sampling time) and tSTS ~ 103 sec (maximum length of MCMC simulation). The distribution also has constant standard deviation in the semilog-x plot, as expected from the analytical calculation of sections S2.1

Fig.S5 (a) µIT and σIT increases with tSTS as power-law (with time exponent ~0.25 for µIT and ~0.1 for σIT), whereas γIT reduces with tSTS. Reduction in γIT indicates that f(NIT) is approaching towards a normal distribution at longer tSTS. (b) The normal probability plot also confirms the existence of positive skew even at tSTS ~ 900sec for Fig. S3b

Fig. S6 Both PDF and CDF of NIT@tSTS~ 900sec can be consistently explained using the skew-normal distribution of eqns. (14)-(15). Other statistics (Gaussian, Poisson, and log-normal) are also fitted with the simulated data using maximum likelihood estimation (MLE). Sum of square errors (SSE) for different distribution functions are: 4.6x105 for skew-normal (which is the lowest of all the fittings), 4x10-4 for Gaussian, 11.4x10-3 for Poisson, and 2x10-3 for log-normal